\magnification = 2200 %\magstep3
%\vsize=1.05\vsize

\def\UseTimesRoman{
\font\cmr=Times
\font\TR=Times at 10pt
\font\TRXII=Times at 12pt
\font\TRXIV=Times at 14pt
\font\TRXX=Times at 20pt
\font\TRXXIV=Times at 24pt
\font\TI=TimesI at 10pt     %Times Italic
\font\TB=TimesB at 10pt     %Times Bold
\font\TBI=TimesBI at 10pt   %Times Bold Italic
\font\TBIviii=TimesBI at 8pt
\font\TBIv=TimesBI at 5pt
%\font\TO=TimesO at 10pt  %Times Oblique (Times Roman, slanted 22%
  %    with EdMetrics)
\font\TO=TimesI at 10pt  %Times Oblique (Times Roman, slanted 22% with EdMetrics)

\font\TIVIII=TimesI at 8pt
\font\TRVIII=Times at 8pt
\font\TIVI=TimesI at 6pt
\font\TRVI=Times at 6pt

	     \font\tenrmscld=Times at 10 pt
        \font\sevenrmscld=Times at 7 pt
        \font\fivermscld=Times at 5 pt

        \font\teniscld=cmmi10 at 10.3 pt
        \font\seveniscld=cmmi10 at 7.21 pt
        \font\fiveiscld=cmmi10 at 5.15 pt
        \font\tensyscld=cmsy10 at 10.3 pt
        \font\sevensyscld=cmsy10 at 7.21 pt
        \font\fivesyscld=cmsy10 at 5.15 pt
        \font\tenexscld=cmex10 at 10.3 pt
        \font\tenbfscld=cmbx10 at 10.3 pt
        \font\sevenbfscld=cmbx10 at 7.21 pt
        \font\fivebfscld=cmbx10 at 5.15 pt

\font\Courier = Courier
\font\Symbol = Symbol

\def\Omega{\hbox{{\Symbol W}}}

\textfont0=\tenrmscld \scriptfont0=\sevenrmscld\scriptscriptfont0=\fivermscld
\def\rm{\fam0\tenrmscld}
\textfont1=\teniscld \scriptfont1=\seveniscld \scriptscriptfont1=\fiveiscld
\def\mit{\fam1} \def\oldstyle{\fam1\teni}
\textfont2=\tensyscld \scriptfont2=\sevensyscld \scriptscriptfont2=\fivesyscld
\def\cal{\fam2}
\textfont3=\tenexscld \scriptfont3=\tenexscld \scriptscriptfont3=\tenexscld
\def\it{\TI}
\def\sl{\TO}
\def\bf{\TB}
\def\rm{\TR}
%\def\tt{\ttCourier}
\def\tt{\Courier}
\def\abstractfont{\TRVIII}
\def\footnotefont{\TRVIII}
\def\tinyfont{\TRvi}
\def\smalltitlefont{\TRXII}
\def\titlefont{\TRXIV}
\def\bigtitlefont{\TRXX}
\def\verybigtitlefont{\TRXXIV}
\textfont9=\TBI \scriptfont9=\TBIviii \scriptscriptfont9=\TBIv
\def\mbi{\fam9}
\rm
        }

%\UseTimesRoman

\def\BR{\Bbb R}             % Besondere Buchstaben
\def\BC{\Bbb C}
\def\BI{\Bbb I}
\def\BN{\Bbb N}
\def\BQ{\Bbb Q}
\def\BS{\Bbb S}
\def\BZ{\Bbb Z}
\def\Tilde{$_{\hbox{\cmrXX \~{}}}$}
\def\ST{\hbox{\eu T }}
\def\SRS{\hbox{\eu RS}}
\def\i{\hbox{{\bf i}}}

\font\sc=cmcsc10 at 10 pt   %% or: at 10 pt
\font\eu=eusb10 %at 10 pt
\font\small=cmr8 at 8 pt
\font\cmrX=cmbx10 scaled \magstep 1 %% 12 point CM
\font\cmrXX=cmbx12 scaled \magstep 1 %%

\hsize 7 true in
\vsize 9 true in
\hoffset = -0.20 true in
\voffset -0.25 true in
\parskip=3pt

\overfullrule = 0pt



\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input epsf.def% \input epsf %for UNIX
%\input epsf          %\input epsf.def for MAC f"ur BILDER!!
\input pics.tex

\input BoxedEPS
\SetTexturesEPSFSpecial
\HideDisplacementBoxes

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\Lf{\vskip1pt\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\R90{{\rm Rot}(90^\circ)}
\def\Dd#1{{\partial \over \partial #1}}

\nopagenumbers

\vglue -10pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cl {\bf About  Hilbert's Cube Filling Curve }
\lf
\cl{See also the more famous Hilbert SquareFillCurve.}
\Lf
Hilbert's cube filling curve is a continuous curve whose image fills a cube. It
is a straight forward generalization of the continuous square filling curve. It
is shown in anaglyph stereo
via a sequence of polygonal approximations. Each approximation
is a polygon that joins two neighboring vertices of the cube.
\lf
The iteration step goes as follows: 
\lf
The cube with the given (initial or a later) approximation is scaled with the
factor $1/2$. Eight of these smaller copies are put together so that they again 
make up the original cube, and this is done in such a way that the endpoint
of the curve in the first cube and the initial point of the curve in the second
cube fit together, and so on with all eight cubes. The result of one iteration
therefore is a curve that is four times as long as the previous curve and
that runs more densely through the cube. In 3DXM, if one rotates the cube
with the mouse then the cube and its first subdividing eight cubes are shown
together with one iteration of the initial curve.
\Lf
To acheive a better feeling for the iteration step, one can set the parameter 
cc to integer values between 0 and 5. This will select different initial curves. 
An even value of cc and the following odd value give
the same initial curve, but for even cc the Hilbert iteration is done 
{\it without} the endpoints, while for odd cc the endpoints are {\it included} in
the iteration. (Using the Action Menu, one can switch between Hilbert's default
(cc=0) and a case that emphasizes the iteration of the endpoints, cc=5.)
\lf
We have the same situation as in the two-dimensional case: The endpoints and 
their iterates are points that already lie on the limit curve because they are
not changed under further iterations. One can say that the endpoints and
their iterates are related to the limit curve in a very simple way. On the other
hand, the approximating polygons develop double points at these iterates
and the result is that the approximations look much more confusing if the
endpoints and their iterates are included in the iteration. This is why we offer
the choice between iterating with and without the endpoints.



\bye
 

 